Showing papers on "Similarity solution published in 1987"
TL;DR: In this article, a model for treating the transfer of angular momentum within a gaseous differentially rotating disc subject to gravitational instability is discussed in terms of an effective kinematic viscosity.
Abstract: A model for treating the transfer of angular momentum within a gaseous differentially rotating disc subject to gravitational instability is discussed in terms of an effective kinematic viscosity. It is assumed that even when matter in the disc is subject to self-gravitation, the instability does not necessarily lead directly to condensation of parts of the disc into self-gravitating bodies. Conditions under which the present model permits a similarity solution are discussed, and it is shown that the general solution tends to the similarity solution at large times.
333 citations
TL;DR: In this article, a similarity solution method was proposed to provide very accurate solutions for laminar forced convection heat transfer from either an isothermal surface or a uniform-flux boundary to fluids of any Prandtl number.
156 citations
TL;DR: In this paper, the relationship of axisymmetric flow between large but finite coaxial rotating disks to the von Karman similarity solution is studied by means of a combined asymptotic- numerical analysis.
Abstract: The relationship of the axisymmetric flow between large but finite coaxial rotating disks to the von Karman similarity solution is studied. By means of a combined asymptotic – numerical analysis, the flow between finite disks of arbitrarily large aspect ratio, where the aspect ratio is defined as the ratio of the disk radii to the gap width separating the disks, is examined for two different end conditions: a ‘closed’ end (shrouded disks) and an ‘open’ end (unshrouded or free disks). Complete velocity and pressure fields in the flow domain between the finite rotating disks, subject to both end conditions, are determined for Reynolds number (based on gap width) up to 500 and disk rotation ratios between 0 and – 1. It is shown that the finite-disk and similarity solutions generally coincide over increasingly smaller portions of the flow domain with increasing Reynolds number for both end conditions. In some parameter ranges, the finite-disk solution may not be of similarity form even near the axis of rotation. It is also seen that the type of end condition may determine which of the multiple similarity solutions the finite-disk flow resembles, and that temporally unstable similarity solutions may qualitatively describe steady finite-disk flows over a portion of the flow domain. The asymptotic – numerical method employed has potential application to related rotating-disk problems as well as to a broad class of problems involving flow in regions of large aspect ratio.
112 citations
TL;DR: In this paper, the problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined, and the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter are derived.
Abstract: The problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream is examined. It was previously shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter.
106 citations
TL;DR: In this article, a similarity solution for a natural convection flow on a heated isothermal wall suspended in a quiescent, thermally stratified atmosphere was obtained for the case of an isothermal plate in a uniform temperature, quiescence medium.
67 citations
TL;DR: In this article, the problem of free convective heat transfer from a nonisothermal ar axisymmetric body of arbitrary geometric configuration in a fluid-saturated porous medium was analyzed on the basis of boundary layer approximations.
Abstract: The problem of free convective heat transfer from a nonisothermal two-dimensional ar axisymmetric body of arbitrary geometric configuration in a fluid-saturated porous medium was analyzed on the basis of boundary layer approximations. Upon introducing a similarity variable (which also accounts for a possible wall temperature effect on the boundary layer length scale), the governing equations for a nonisothermal body of arbitrary shape can be reduced to an ordinary differential equation which has been previously solved by Cheng and Minkowycz for a vertical flat plate with its wall temperature varying in an exponential manner. Thus, it is found that any two-dimensional or axisymmetric body possesses a corresponding class of surface wall temperature distributions which permit similarity solution. Furthermore, a more straightforward and yet sufficiently accurate approximate method based on the Karman-Pohlhausen integral relation is suggested for a general solution procedure for a Darcian fluid flow over a nonisothermal body of arbitrary shape. For illustrative purposes, computations were carried out on a vertical flat plate, horizontal ellipses, and ellipsoids with different minor-to-major axis ratios.
60 citations
TL;DR: In this article, the authors characterize the most general 2 × 2 first order quasilinear hyperbolic systems in conservative form which are invariant with respect to the stretching group of transformation.
Abstract: We characterize the most general 2 × 2 first order quasilinear hyperbolic systems in conservative form which are invariant with respect to the stretching group of transformation. The invariant solutions satisfy, under suitable conditions, an autonomous first order system of ordinary differential equations. A procedure is given to characterize the profile of the velocities of the strong and weak discontinuities.
47 citations
TL;DR: In this paper, the similarity solution of two-dimensional point vortices in an unbounded region is presented, and the time dependence of the solution for all vortex positions are assumed to be the same.
Abstract: Theory of the similarity solution of two-dimensional point vortices in an unbounded region is presented. To get the solution, the time dependence of the solution for all vortex positions are assumed to be the same. It is shown that the system either rotates rigidly or collapses according to the initial position of vortices. Then numerical calculations are made especially for the system of three vortices. It is found that the rigid rotation corresponds to the regular triangle solution or the straight line solution, and that irrespective of the strengths of three vortices, the regular triangle solution always exists while the straight line solution is obtained by solving a certain cubic equation.
45 citations
TL;DR: In this paper, the Navier-Stokes equations were matched at the interface and it was found that the flow field depends heavily on the ratios of viscosity and density.
Abstract: Jets of different fluid properties impinge head‐on. The region near the stagnation point is investigated. Similarity solutions of the Navier–Stokes equations are matched at the interface. It is found that the flow field depends heavily on the ratios of viscosity and density.
33 citations
TL;DR: In this paper, the authors match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock-wedge diffraction problem.
Abstract: We match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock-wedge diffraction problem.
30 citations
TL;DR: In this paper, the mixed convection boundary layer on a horizontal plate is considered for two separate cases when there is a uniform free stream with the plate held fixed and when there was no outer flow but the plate was moving continuously with a uniform velocity along its length.
Abstract: The mixed convection boundary layer on a horizontal plate is considered for the two separate cases when there is a uniform free stream with the plate held fixed and when there is no outer flow but the plate is moving continuously with a uniform velocity along its length. In both cases we assume that power law temperature distribution on the plate which enables the governing equations to be reduced to similarity form. For the first problem we consider the range of buoyancy parameter for which there are dual solutions, showing how these dual solutions arise from a bifurcation and how the lower branch of solutions terminate as the buoyancy parameter tends to zero. For the second problem we show that there is a unique solution for all positive values of the buoyancy parameter and that for negative values the solution terminates at a singular solution with algebraic decay.
TL;DR: In this paper, an exact similarity solution for a spherical magnetogasdynamic shock is obtained in the case when radiation energy, radiation pressure and radiative heat flux are important, and the total energy of the shock wave increase with time.
Abstract: An exact similarity solution for a spherical magnetogasdynamic shock is obtained in the case when radiation energy, radiation pressure and radiative heat flux are important. The total energy of the shock wave increase with time. We have shown that due to the magnetic field the flow variables are considerably changed. Also, due to increases in radiation pressure number the radiation flux is increased.
TL;DR: In this paper, a similarity analysis of the non-linear two-dimensional non-stationary ideal MHD equations is presented and the complete Lie algebra of infinitesimal symmetries is established in the case of a magnetic field perpendicular to the isentropic motion of the plasma.
Abstract: A similarity analysis of the non-linear two-dimensional non-stationary ideal MHD equations is presented. In the case of a magnetic field perpendicular to the isentropic motion of the plasma, the authors establish the complete Lie algebra of infinitesimal symmetries. The laws of conservation are mentioned. The similarity method for partial differential equations as a procedure for reducing the number of independent variables is applied repeatedly. Finally, they obtain systems of ordinary differential equations for similarity solutions of the MHD equations considered.
TL;DR: In this article, a two-dimensional model of a constant-density diffusion-controlled reaction between unmixed species initially occupying adjacent half-spaces is formulated and analyzed, and a global similarity solution to the problem is Fourier analyzed in a Lagrangian coordinate system.
Abstract: A two-dimensional model of a constant-density diffusion-controlled reaction between unmixed species initially occupying adjacent half-spaces is formulated and analyzed. An axisymmetric viscous vortex field satisfying the Navies-Stokes equations winds up thc interface between the species as they diffuse together and react. A flame-sheet approximation of the rapid reaction is made using a mixture fraction dependent variable. The problem was originally proposed by F. Marble, who performed a local analysis and determined the total consumption rate along thc flame sheet. The present paper describes a global similarity solution to the problem which is Fourier analyzed in a Lagrangian coordinate system. The Fourier amplitudes are determined both by an asymptotic analysis, valid for large Schmidt numbers,and by numerical solution of the two-point boundary-value ordinary differential equations. The solution is evaluated in both Lagrangian and Eulerian coordinate systems. Comparisons are made between the a...
TL;DR: In this paper, a similarity solution of the Leslie-Ericksen equations for nematic liquid crystals is obtained for converging and diverging flow, which shows a boundary layer resulting from interactions between wall and flow-induced orientations; the boundary layer scaling with Ericksen number differs, depending on whether the interaction is with the shear flow near the wall or the extensional flow near channel midplane.
Abstract: A similarity solution of the Leslie-Ericksen equations for nematic liquid crystals is obtained for converging and diverging flow. The director distribution for parameters characteristic of lyotropic liquid crystal polymer solutions shows a boundary layer resulting from interactions between wall- and flow-induced orientations; the boundary layer scaling with Ericksen number differs, depending on whether the interaction is with the shear flow near the wall or the extensional flow near the channel midplane. Imposition of an azimuthal magnetic field causes a first-order Freedericksz-like transition in director orientation at a critical field strength.
TL;DR: A coupled system of nonlinear evolution equations describing the combined effects of diffusion and reactions, e.g., fusion reactions or electron-ion recombinations, is transformed into a system of coupled similarity equations in one, two, and three dimensions.
Abstract: A coupled system of nonlinear evolution equations describing the combined effects of diffusion and reactions, e.g., fusion reactions or electron-ion recombinations, is transformed into a system of coupled similarity equations in one, two, and three dimensions. A description of the evolution of the coupled system for special though physically relevant situations, e.g., the diffusion of an ignited high-density fusion plasma (thermal-thermal reactions) of tritium and deuterium, is obtained in terms of a particular similarity solution.
TL;DR: In this article, the effect of an external field on dilute suspensions of dipolar axisymmetric Brownian particles in a Newtonian solvent is considered and a family of similarity solutions is derived for the orientation distribution of particles after inception of steady two-dimensional flow in the plane normal to the field.
Abstract: We consider the effect of an external field on dilute suspensions of dipolar axisymmetric Brownian particles in a Newtonian solvent. A family of similarity solutions is derived for the orientation distribution of particles after inception of steady two-dimensional flow in the plane normal to the field. It is assumed that the particles are initially aligned by the field. The solution is uniformly valid for small times, but if the field is strong enough to overcome diffusion, the solution remains valid at all time, correctly predicting the steady state distribution. The rheological properties are obtained in closed form from the similarity solution and the role of the external field is demonstrated. First and second normal stress differences are obtained. The solutions are presented for particles with fixed dipoles, although they apply equally to particles with dipoles induced by the external field.
01 Jun 1987
TL;DR: In this article, a fully conservative, viscous, implicit, upwind, finite-volume scheme for the thin-layer Navier-Stokes equations is described with application to blast wave flow fields.
Abstract: A fully conservative, viscous, implicit, upwind, finite-volume scheme for the thin-layer Navier-Stokes equations is described with application to blast wave flow fields. In this scheme, shocks are captured without the oscillations typical of central differencing techniques and wave speeds are accurately predicted. The finite volume philosophy ensures conservation and since boundary conditions are also treated conservatively, accurate reflections of waves from surfaces are assured. Viscous terms in the governing equations are treated in a manner consistent with the finite volume philosophy, resulting in very accurate prediction of boundary layer quantities. Numerical results are presented for four viscous problems: a steady boundary layer, a shock-induced boundary layer, a blast wave/cylinder interaction and a blast wave/supersonic missile interaction. Comparisons of the results with an established boundary layer code, similarity solution, and experimental data show excellent agreement.
TL;DR: Similarity solutions for line explosion in a non-uniform self-gravitating medium including the effects of magnetic field radiation flux and neglecting the radiation pressure and energy are investigated in this paper.
Abstract: Similarity solutions for line explosion in a non-uniform self-gravitating medium including the effects of magnetic field radiation flux and neglecting the radiation pressure and energy are investigated. Gas is assumed to be grey and opaque and the shock to be transparent and isothermal.
TL;DR: In this article, a new similarity variable is proposed to solve the problems of steady film condensation and boiling over a body of arbitrary shape embedded in a porous medium. But the similarity variable does not solve the problem for all possible two-dimensional and axisymmetric bodies of arbitrary geometrical configuration.
TL;DR: In this paper, the authors describe methods for modeling the rate of growth of solid in terms of a mass exchange, incorporating drag, buoyancy and interfacial pressure forces in the momentum exchange, and including the latent heat and dissipation in the energy exchange.
TL;DR: In this article, the phase trajectories of particles in a plasma described by the one-dimensional Vlasov-Maxwell equations are determined qualitatively, analyzing exact general similarity solutions for the cases of temporally damped and growing (sinusoidal or localized) electric fields.
Abstract: The phase trajectories of particles in a plasma described by the one-dimensional Vlasov-Maxwell equations are determined qualitatively, analyzing exact general similarity solutions for the cases of temporally damped and growing (sinusoidal or localized) electric fields. The results of numerical integration in both untransformed and Lie-group point-transformed coordinates are presented in extensive graphs and characterized in detail. The implications of the present analysis for the stability of BGK equilibria are explored, and the existence of nonlinear solutions arbitrarily close to and significantly different from the BGK solutions is demonstrated.
TL;DR: The analysis of free convection heat transfer near a corner formed by two mutually perpendicular flat plates embedded in porous medium is presented in this article, where the authors find that local similarity solution exists for the case of constant wall temperature.
TL;DR: In this article, the Lie point symmetries of a set of coupled nonlinear partial differential equations are considered, and the system is an extended version of the usual nonlinear Schrodinger equation.
Abstract: The Lie point symmetries of a set of coupled nonlinear partial differential equations are considered. The system is an extended version of the usual nonlinear Schrodinger equation. In the similarity variable deduced from the symmetry analysis, the system is equivalent to the Painleve III in Ince's classification. By starting from a solution of the Painleve equation, one can reproduce various classes of solutions of the original PDEs. Such solutions include both rational and progressive types or a combination of the two.
01 Apr 1987
TL;DR: In this article, the global method of Bridges and Morris was chosen for the spatial stability calculations for the nonlinear eigenvalue problem, and a second order accurate central difference scheme was used to obtain the coefficient matrices.
Abstract: A brief overview of some theoretical and computational studies of the stability of streamwise vortices is given. The local induction model and classical hydrodynamic vortex stability theories are discussed in some detail. The importance of the three-dimensionality of the mean velocity profile to the results of stability calculations is discussed briefly. The mean velocity profile is provided by employing the similarity solution of Donaldson and Sullivan. The global method of Bridges and Morris was chosen for the spatial stability calculations for the nonlinear eigenvalue problem. In order to test the numerical method, a second order accurate central difference scheme was used to obtain the coefficient matrices. It was shown that a second order finite difference method lacks the required accuracy for global eigenvalue calculations. Finally the problem was formulated using spectral methods and a truncated Chebyshev series.
TL;DR: In this paper, Lie's infinitesimal transformation groups, which leave the basic equations of axially symmetric problems of classical elasticity invariant, are constructed for the case of the Boussinesque's problem of an elastic semi-space subjected to a point force applied normal to its surface.
Abstract: Lie's infinitesimal transformation groups, which leave the basic equations of axially symmetric problems of classical elasticity invariant, are constructed. For the case of the axisymmetric Boussinesque's problem of an elastic semi-space subjected to a point force applied normal to its surface, the invariance of boundary and boundary conditions leads to the explicit form of similarity transformations which are used to solve the problem. Expressions for the displacements and stresses derived by this approach, which is believed to be new, are found to agree with the known results.
TL;DR: In this paper, numerical solutions for the similarity formulation of the equations, closed with an eddy viscosity model, governing the flow in a buoyant plume in a stratified environment are found.
Abstract: Numerical solutions are found for the similarity formulation of the equations, closed with an eddy viscosity model, governing the flow in a buoyant plume in a stratified environment The effects of the stratification on the velocity and temperature profiles are studied Comparisons between computed results and experimental data are made
TL;DR: In this paper, the round laminar plume issued in a stratified environment has been studied under the boundary layer approximation, and the differential equations governing the problem have been solved numerically using the finite difference method developed by Patankar-Spalding.